lower bounds for perron root of positive matrices

S Ü Fen Ed Fak Fen Derg
Sayı 22 (2003) 71- 76, KONYA
On the Upper Bounds for Permanents
Ahmet Ali ÖÇAL1
Abstract: In this paper, considering
upper bounds for permanents.
λ1 , λ 2
and λ ∞ operator norms, we obtained some
Key Words: λ1 , λ 2 and λ ∞ operator norms, permanent
Permanentlerin Üst Sınırları Üzerine
Özet: Bu çalışmada, λ1 , λ 2 ve λ ∞ operatör normları gözönüne alınarak permanentler
için bazı üst sınırlar elde edilmiştir.
Anahtar Kelimeler: λ1 , λ 2 ve λ ∞ operatör normları, permanent
Introduction and the Statemens of Results
Definition 1. [1] The permanent of a real n×n matrix A = (a ij ) is defined by
per ( A ) =
n
∑∏ a
σ∈S n i =1
iσ( i )
,
where S n is the symmetric group of order n.
Definition 2. ([2]) The λ1 operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined
{
}
A 1 = max Ax 1 : x ∈ Cn , x 1 = 1 ,
where x = ( x1 , x 2 ,Κ , x n )T , (T denoting the transpoze) and
n
x 1 = ∑ xi .
i =1
Definition 3. ([2]) The λ2 operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined
A
1
2
{
= max Ax 2 : x ∈ Cn , x
Department of Mathematics, University of Gazi, 42500 Ankara, TURKEY
2
}
=1 ,
On the Upper Bounds for Permanents
where x = ( x1 , x 2 ,Κ , x n )T and
1
x
 n
2 2
=  ∑ x i  .

 i =1
2
Definition 4. ([2]) The λ∞ operator norm of an n×n matrix A = (aij ) ∈ Cn×n is defined
A
∞
{
= max Ax ∞ : x ∈ Cn , x
∞
}
=1 ,
where x = ( x1 , x 2 ,Κ , x n )T and
x
∞
= max x i .
1≤i≤n
Lemma 1. Let a1 , a2 ,Κ , an be the columns of A = (aij ) ∈ Cn×n . Then
( n)
n
per ( A ) ≤
a1
a2
2
2
Κ an
2
,
where
aj
2

=


1
n
∑a
ij
i =1
2 2



, 1≤ j ≤n.
Proof. We make use of the inequality (see e.g. [1, p.113])
n
per ( A ) ≤ ∏ c i
i=1
where c 1 , c 2 ,Κ , c n are column sums of A and A = (aij )n×n is a nonnegative matrix. Since
per ( A ) ≤ per ( A )
by the triangle inequality, any such bound can be used to produce an upper bound for the
permanents of complex matrices. For example from the inequality (1), we obtain
n
per ( A ) ≤ ∏ q j ,
j=1
where
n
q j = ∑ a ij , j = 1, 2 ,Κ , n .
i=1
By the Cauchy-Schwarz Inequality, we have
72
(2)
Ahmet Ali ÖÇAL
n
qj =
∑
i =1

aij ≤ 


1
n
∑a
ij
i =1
2 2

 
 
 
1
2 
1 =


i =1 

n
∑
1
2 2
n
∑a



ij
i =1
n = aj
2
n.
So from inequality (2) we obtain
per ( A ) ≤
( n)
n
a1
2
a2
2
Κ an
2
and the proof is complete.
Theorem 1. Let
A
2
{
= max Ax 2 : x ∈ C n , x
2
=1
}
be λ2 operator norm of A ∈ C n×n . Then
n
n
per ( A ) ≤ n 2 A 2 .
Proof. Denote the columns of A by a1 , a2 ,Κ , an and let e1 , e 2 ,Κ , e n be the standart
basis of Cn . Then we have
aj = A ej , 1≤ j ≤ n .
(3)
So considering Lemma 1 we have
n
per ( A ) ≤ n 2 a1
a 2 2 Κ an
2
n
2


≤ n  max a j 
2

 1≤ j≤n
2
n
n
2


= n  max Ae j 
2
 1≤ j≤n



≤ n  max Ax 2 
 x 2 =1

n
2
=n
n
2
A
n
n
n
2
and thus the theorem is proved.
Lemma 2. Let a1 , a2 ,Κ , an be the columns of A ∈ C n×n . Then
73
On the Upper Bounds for Permanents
n
per ( A ) ≤ ∏ a j ,
1
j=1
where
aj
n
1
= ∑ a ij , j = 1, 2 ,Κ , n .
i=1
Proof. The proof of Lemma is immediately seen from (2).
Theorem 2. Let
{
A 1 = max Ax 1 : x ∈ Cn , x 1 = 1
}
be λ1 operator norm of A ∈ C n×n . Then
n
per ( A ) ≤ A 1 .
Proof. Considering Lemma 2 and the equality (3), we have
per ( A ) ≤ a1
1
a 2 1 Κ an
≤ Ae1 1 Ae 2 1 Κ Aen

≤  max Ae j
 1≤ j≤n


1



≤  max Ax 1 
 x =1

 1

= A
n
1
1
1
n
n
.
Thus the theorem is proved.
Lemma 3. Let a1 , a2 ,Κ , an be the columns of A = (a ij ) ∈ C n×n . Then
aj
1
≤ n aj
∞
,
where
aj
74
n
1
= ∑ a ij , j = 1, 2 ,Κ , n ,
i=1
Ahmet Ali ÖÇAL
and
aj
= max a ij , 1 ≤ j ≤ n .
∞
1≤i≤n
Proof. For all j, 1 ≤ j ≤ n , we have
aj
n
1
= ∑ a ij = a1j + a 2 j + Λ + a nj
i=1
≤ n max a ij
1≤i≤n
= n aj
∞
.
Thus the proof is complete.
Theorem 4. Let a1 , a2 ,Κ , an be the columns of A = (a ij ) ∈ C n×n . Then
n
per ( A ) ≤ n n ∏ a j
j=1
∞
.
Proof. Considering Lemma 2 and Lemma 3 the proof is easily seen.
Theorem 5. Let
A
∞
{
= max Ax ∞ : x ∈ Cn , x
∞
=1
}
be λ∞ operator norm of A .Then
per ( A ) ≤ n n A
n
∞
.
Proof. From Theorem 4 and equality (3), we have
per ( A ) ≤ n n a1
∞
a2
∞
Κ an
∞
75
On the Upper Bounds for Permanents

≤ n  max a j
 1≤ j≤n
n
∞

= n n  max Ae j
 1≤ j≤n

≤ n  max Ax
 x =1
 ∞
n
= nn A
n
∞




n


∞



∞

n
n
,
and thus the theorem is proved.
REFERENCES
1. Minc, H., Permanents, In Encyclopedia of Mathematics and Its Applications Vol. 6, Addison-Wesley (1978).
2. Taşcı, D., On a Conjecture by Goldberg and Newman, Linear Algebra and its Appl., 215: 275-277 (1995).
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